These data are required to satisfy a long list of

These data are required to satisfy a long list of coherence diagrams, see [ 55 , Ap-
pendix C] for details.
Definition A.24. A symmetric monoidal 2-functor between two symmetric monoidal
bicategories B and B
0
consists of a 2-functor H : B -→ B
0
of the underlying bicat-
egories together with the following data:
(a) Equivalence natural 2-transformations 3 χ : ⊗ 0 ◦
(
H ( · ) × H ( · )
)
= ⇒ H ◦ ⊗ and
ι : 1
0
= ⇒ H (1) , where here we consider 1 as a 2-functor from the bicategory
with one object, one 1-morphism and one 2-morphism to B .
(b) The three invertible modifications
H ( a ) ⊗
0
(
H ( b ) ⊗
0
H ( c )
)
H ( a ) ⊗
0
H ( b ⊗ c )
(
H ( a ) ⊗
0
H ( b )
)
⊗
0
H ( c ) H
(
a ⊗ ( b ⊗ c )
)
H ( a ⊗ b ) ⊗
0
H ( c ) H
(
( a ⊗ b ) ⊗ c
)
id ⊗
0
χ
χ
α
0
χ ⊗
0
id
χ
H ( α )
Ω
(A.25)
3 We fix again adjoint inverses and the adjunction data.
152

Chapter A: Symmetric monoidal bicategories
H (1) ⊗ 0 H ( a ) H (1 ⊗ a )
1
0
⊗
0
H ( a ) H ( a )
χ
H ( λ )
ι ⊗
0
id
λ
0
Γ
and
H ( a ) ⊗ 0 1 0 H ( a ) ⊗ 0 H (1)
H ( a ) H ( a ⊗ 1)
id ⊗
0
ι
χ
ρ
0
H ( ρ )
Δ
(c) An invertible modification
H ( b ⊗ a )
H ( b ) ⊗ 0 H ( a ) H ( a ⊗ b )
H ( a ) ⊗
0
H ( b )
H ( β )
Υ
χ
β
0
χ
(A.26)
These data are required to satisfy a long list of coherence conditions, see [ 55 ] and
references therein for details.
Our definition of symmetric monoidal transformations differs slightly from [ 55 ].
The definition we give is tailored to the application in functorial field theories. In
contrast to the definition given in [ 55 ], we require the appearing modifications to
be invertible. However, the 2-morphisms corresponding to the underlying natural
transformations are not invertible in our definition, so our definition is also weaker
than the definition given in [ 55 ].
Definition A.27. A natural symmetric monoidal 2-transformation between sym-
metric monoidal 2-functors H , K : B -→ B
0
consists of a natural 2-transformation
θ : H = ⇒ K of the underlying 2-functors together with invertible modifications
H ( a ⊗ b )
H ( a ) ⊗
0
H ( b ) K ( a ⊗ b )
K ( a ) ⊗ 0 H ( b ) K ( a ) ⊗ 0 K ( b )
θ
χ
H
θ ⊗
0
id
id ⊗
0
θ
Π
χ
K
(A.28)
and
1
0
K (1)
H (1)
ι
K
ι
H
M
θ
(A.29)
153

Chapter A: Symmetric monoidal bicategories
which satisfy the following coherence conditions expressed as equalities between 2-
morphisms (omitting tensor product symbols on objects and 1-morphisms to stream-
line the notation):
K ( a )
(
K ( b ) H ( c )
)
K ( a )
(
K ( b ) K ( c )
) (
( K ( a ) K ( b )
)
K ( c )
(
K ( a ) K ( b )
)
H ( c ) K ( ab ) K ( c )
K ( ab ) H ( c )
(
K ( a ) H ( b )
)
H ( c ) K
(
( ab ) c
)
H ( ab ) H ( c )
(
H ( a ) H ( b )
)
H ( c ) H
(
( ab ) c
)
K
(
a ( bc )
)
H ( a )
(
H ( b ) H ( c )
)
H ( c ) H ( bc ) H ( a ( bc ))
θ α
0
χ
K
α
0
θ
χ
K
χ
K
θ
Π
θ
Π ⊗
0
id
K ( α )
θ
χ
H
Ω
H
θ
α
θ
χ
H
α
0
H ( α )
θ
χ
H
χ
H
θ
k
K ( a )
(
K ( b ) H ( c )
)
K ( a )
(
K ( b ) K ( c )
) (
K ( a ) K ( b )
)
K ( c )
(
K ( a ) K ( b )
)
H ( c ) K ( a )
(
K ( b ) K ( c )
)
K ( ab ) K ( c )
(
K ( a ) H ( b )
)
H ( c ) K ( a ) K ( bc ) K
(
( ab ) c
)
K ( a )
(
H ( b ) H ( c )
)
(
H ( a ) H ( b )
)
H ( c ) K ( a ) H ( bc ) K
(
a ( bc )
)
H ( a )
(
H ( b ) H ( c )
)
H ( a ) H ( bc ) H
(
a ( bc )
)
θ
id ⊗
0
Π
α
0
χ
K
χ
K
α
0
α
0
χ
K
χ
K
Ω
K
α
0 ?
θ
α
0
χ
K
Π
K ( α )
θ
Φ
⊗
0
χ
H
θ
α
0
α
0 ?
θ
χ
H
θ
θ
χ
H
θ
(A.30)
154

Chapter A: Symmetric monoidal bicategories
K (1) H ( a ) K (1) K ( a )
H (1) H ( a ) H (1 a ) K (1 a )
1 0 H ( a ) H ( a ) K ( a )
1 0 K ( a )
θ
Π
χ
K
θ
χ
H
θ
H ( λ ) K ( λ )
ι
H
λ
0
θ
θ
λ
0
θa
λ
0
Γ
H
θ
k
K (1) H ( a ) K (1) K ( a )
H (1) H ( a ) K (1 a )
1 0 H ( a ) K ( a )
1
0
K ( a )
θ
Φ
⊗
0
χ
K
θ
M
- 1
⊗
0
id
K ( λ )
ι
H
ι
K
θ
Γ
K
λ
0
ι
K
(A.31)
K ( a )1 0 K ( a ) H (1)
K ( a ) H ( a )1 0 H ( a ) H (1) K ( a ) K (1)
H ( a ) H ( a 1) K ( a 1)
K ( a )
ι
H
θ
ρ
0
ρ 0
θ
θ
ι
H
Π
χ
H
θ
χ
K
θ
θ
ρ 0
H ( ρ )
Δ
- 1
H
θ
ρ
θ
K ( ρ )
ι
θ
k
155

Chapter A: Symmetric monoidal bicategories
K ( a )1 0 K ( a ) H (1)
K ( a ) K ( a ) K (1)
H ( a ) K ( a 1)
K ( a )
ι
K
ι
H
θ
ρ
0
K ( ρ )
id
id ⊗
0
M
- 1
χ
K
θ
θ
K ( ρ )
Δ
- 1
K
(A.32)
and
H ( b ) H ( a ) H ( ba )
H ( b ) H ( a ) H ( ab ) K ( ba )
K ( a ) K ( b ) K ( ab )
χ
H
Υ